Conversely, let us prove that \(Age(P_\infty )\)  and \(Age(H)\)  are minimal prime. This is obvious for \(Age(P_\infty )\) , indeed, finite prime subgraphs of \(P_{\infty }\)  on at least three vertices are paths, thus form a chain for the embeddability order; if a hereditary subclass of \(Age(P_{\infty })\)  contains infinitely many of these paths it contains all, thus is equal to \(Age(P_{\infty })\) . For \(Age(H)\) , the situation is about the same: finite prime induced subgraphs of \(H\)  on at least four vertices form also a chain. Indeed, these subgraphs are the critically prime discovered by Schmerl and Trotter [1]}, prime graphs such that the deletion of any vertex yields a nonprimitive graph. As shown by Schmerl and Trotter, these graphs have an even number of vertices; up to complementation, there is just one critically prime graph on an even number of vertices.
 \(\Box \)
